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<p>This paper is devoted to the origins and detailed analysis of the complex bifurcations within micromechanical frequency combs (MFCs) based on a nonlinear MEMS resonator. The flexural and torsional modes of the resonator are designed to a 1:3 frequency ratio, which triggers the primary subcritical Hopf bifurcation via 1:3 internal resonance. The dynamic states distribution is theoretically modeled by a reduced-order coupling system, predicting the exact frequency of Hopf bifurcation and Neimark-Sacker bifurcation through multiple scales methods. To validate the proposed theory, experiments are performed to distinguish the boundaries among vibration-forbidden zones, chaos, periodic motion, and quasiperiodic MFCs. Within the MFCs regime, it is found that the relationship between coupling transferred energy and vibration amplitude influences the number and size of the limit cycles. When the coupling transferred energy cannot support the increasing amplitude, the multi-limit cycles will first shrink and then converge to fewer numbers, leading to a continuous 4:3:2:1 period-doubling bifurcation. When the coupling transferred energy is tuned from strong to weak conditions, the limit cycle expands to its maximum and then contracts to a fixed point, manifesting a cyclic-fold bifurcation. Both experimental and theoretical results of this research work demonstrate that the coupling transferred energy between two modes determines the limit cycles, filling a knowledge gap about the origins of complex bifurcations within MFCs evolution.</p>

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Limit cycle convergence leads to period-doubling and cyclic-fold bifurcation in internal resonance-induced mechanical frequency combs

  • 期刊论文

  • 发表日期：

    2025年4月19日

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  • Jiahao Wu,
  • Penghui Song,
  • Shuke Zang,
  • Lei Li,
  • Wenming Zhang,
  • Lei Shao

摘要

This paper is devoted to the origins and detailed analysis of the complex bifurcations within micromechanical frequency combs (MFCs) based on a nonlinear MEMS resonator. The flexural and torsional modes of the resonator are designed to a 1:3 frequency ratio, which triggers the primary subcritical Hopf bifurcation via 1:3 internal resonance. The dynamic states distribution is theoretically modeled by a reduced-order coupling system, predicting the exact frequency of Hopf bifurcation and Neimark-Sacker bifurcation through multiple scales methods. To validate the proposed theory, experiments are performed to distinguish the boundaries among vibration-forbidden zones, chaos, periodic motion, and quasiperiodic MFCs. Within the MFCs regime, it is found that the relationship between coupling transferred energy and vibration amplitude influences the number and size of the limit cycles. When the coupling transferred energy cannot support the increasing amplitude, the multi-limit cycles will first shrink and then converge to fewer numbers, leading to a continuous 4:3:2:1 period-doubling bifurcation. When the coupling transferred energy is tuned from strong to weak conditions, the limit cycle expands to its maximum and then contracts to a fixed point, manifesting a cyclic-fold bifurcation. Both experimental and theoretical results of this research work demonstrate that the coupling transferred energy between two modes determines the limit cycles, filling a knowledge gap about the origins of complex bifurcations within MFCs evolution.